Since \(G\)  admits an injective holomorphic homomorphism to the group of invertible elements of a Banach algebra, it is linear [1]}. We have by Theorem REF  that \(\ell _\pi \)  is exponentially distorted on the nilpotent radical \(N\)  and so is \(\ell _\pi ^{sym}\)  by Lemma REF . By the hypothesis, \(\ell ^{\prime }\)  is exponentially distorted on \(N^{\prime }\)  and hence so is \(\ell ^{\prime }+ \ell _\pi ^{sym}\) .
To prove that the distortion of \(\ell ^{\prime }+ \ell _\pi ^{sym}\)  is strictly exponential it suffices to show that the distortion of \(\ell _\pi ^{sym}\)  is strictly exponential. Note that by Lemma REF , \(N^{\prime }\)  is nilpotent, closed and simply connected and so we can apply Proposition REF . Thus all we need to show is that
\(\log (1+|\eta |)\lesssim \ell _\pi ^{sym}(\exp (\eta ))\quad \text{on~$\mathfrak {n}^{\prime }$}.\) 
We prove more: this holds on the whole \(\mathfrak {n}\) .
Since \(\pi \)  is injective, the map between the algebras of holomorphic functions on \({\mathop {\mathrm {GL}}\nolimits }(A)\)  and \(G\)  has dense range. The elements of the corresponding Lie algebras can be identified with continuous \(\mathbb {C}\) -valued derivations on these algebras. The density of the range implies that \(\mathrm {L}_\pi \)  is also injective.
Thus \(|\eta |\!:= \Vert \mathrm {L}_\pi (\eta )\Vert \)  is a norm on \(\mathfrak {n}\) . Fix \(\eta \in \mathfrak {n}\)  and let \(a\!:= \mathrm {L}_\pi (\eta )\) . Since \(A\)  is a PI-algebra, so is the closure of the range of \(\theta \!:U(\mathfrak {g})\rightarrow A\) . Applying
Theorem REF  to this closure we conclude that \(a\)  is nilpotent. It can be easily seen that there is \(m\in \mathbb {Z}_+\)  such that \(1,a,\ldots ,a^{m-1}\)  are linearly independent and \(a^m=0\) .
We can assume that \(\eta \ne 0\) .
Then \(a\ne 0\)  by the injectivity. So by the linear independence and the Hahn-Banach theorem, there is a bounded linear functional \(f\)  on \(A\)  such that \(\langle f, 1 \rangle =0\) , \(\langle f,a\rangle =\Vert a\Vert \)  and \(\langle f,a^k\rangle =0\)  when \(k>1\) .
Hence,
\(\langle f, e^{a}\rangle = \sum _{k=0}^\infty \mathfrak {r}ac{\langle f,a^k\rangle }{k!}=\Vert a\Vert .\) 
Then for every \(\eta \in \mathfrak {n}\)  we have \(|\eta |=\Vert a\Vert \le \Vert f\Vert \,\Vert e^{a}\Vert =\Vert f\Vert \,\Vert \pi (\exp (\eta ))\Vert \)  (the last equality by (REF )) and therefore
\(|\eta |\le \Vert f\Vert \, \exp (\ell _\pi (\exp (\eta )))\le \Vert f\Vert \, \exp (\ell _\pi ^{sym}(\exp (\eta ))).\) 
On the other hand, it follows from the definition of \(\ell _\pi ^{sym}\)  that \(1\le \exp (\ell _\pi ^{sym}(g))\)  for every \(g\)  (since \(1=\Vert \pi (1)\Vert \le \Vert \pi (g)\Vert \,\Vert \pi (g^{-1})\Vert \) ). Combining the bounds for 1 and \(|\eta |\)  we have that
\(1+|\eta |\le (1+\Vert f\Vert ) \exp (\ell _\pi ^{sym}(\exp (\eta )))\) 
and so \(\log (1+|\eta |)\le \ell _\pi ^{sym}(\exp (\eta ))+\log (1+\Vert f\Vert )\)  on \(\mathfrak {n}\) . Thus \(\ell ^{\prime }+ \ell _\pi ^{sym}\)  is strictly exponentially distorted on \(N^{\prime }\) .
